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June, 1987 High Breakdown-Point and High Efficiency Robust Estimates for Regression
Victor J. Yohai
Ann. Statist. 15(2): 642-656 (June, 1987). DOI: 10.1214/aos/1176350366


A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the second stage an M-estimate of the errors scale is computed using residuals based on the initial estimate. Finally, in the third stage an M-estimate of the regression parameters based on a proper redescending psi-function is computed. Consistency and asymptotical normality of the MM-estimates assuming random carriers are proved. A convergent iterative numerical algorithm is given. Finally, the asymptotic biases under contamination of optimal bounded influence estimates and MM-estimates are compared.


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Victor J. Yohai. "High Breakdown-Point and High Efficiency Robust Estimates for Regression." Ann. Statist. 15 (2) 642 - 656, June, 1987.


Published: June, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0624.62037
MathSciNet: MR888431
Digital Object Identifier: 10.1214/aos/1176350366

Primary: 62F35
Secondary: 62J05

Keywords: high breakdown-point , high efficiency , linear model , robust estimation

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 2 • June, 1987
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